"For as the woman is of the man, even so is the man also by the woman; but all things of God." (1 Corinthians 11:12)
We cannot and should not make the blunder of establishing a principle based on a single verse. St. Paul didn’t write oracles or proverbs. He is expressing the flow of his thoughts in a letter. We should always look at the start of the argument and also the conclusion to draw inferences and definitely not missing the flow of thoughts. The creation has a design and that’s what we saw the head (authority) of a woman is man and head (authority) of man is Christ and head (authority) of Christ is God. We should not forget this order. Looks like in the Church at Corinth, women used to have a role in the state of affairs. They used to probably pray aloud, uncover their heads as they took part in worship or spiritual services. We also should take time to understand about the Greek, Roman and Jewish customs which were prevalent during those days when we analyse these passages.
Paul asks the women to cover their heads so that they may understand who their head is. Spiritual services should always be done with the mindset of submission and headship. When we bring an element of pride and independence in spiritual service, it brings us to negative consequences and it doesn’t augur well in a church. The veil was also a sign of marriage for a woman (they used to wear it always). So when a woman who was under the authority of a man removed her veil in worship, it showed she was independent and Paul was bringing to their notice the angels who sinned in Genesis 6. The issue is not about wearing a veil now or not, it's about the heart of submission that we need to have to the established order of God. A woman who serves or participates in the church service specially should be armed with the attitude that she is submitted to the marital covenant and divine order. A man should have his submission to his head who is Christ .A woman is the glory of man. The Church was not a place of disorder where ethnic and civil norms were violated. It's not about who is bigger or greater. As woman as made from man, man is now born of woman, but all things are of God.
For Further Reading:
Galatians 3:28 1 Peter 3:1-6